Which point is a focus of the hyperbola?
A. (-11,-4)
B. (-3,-4)
C. (2,-4)
D. (2,8)
Answer (3)
To determine the focus of a hyperbola, we need its center and the distances to the foci. The points given are A. (-11,-4), B. (-3,-4), C. (2,-4), D. (2,8), and without the hyperbola's details, assessing B (-3,-4) may suggest it could be a potential focus depending on its relation to the center. Accurate identification, however, relies on the specific equation of the hyperbola in question.
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To determine the focus of a hyperbola, you need its equation or at least the center, vertices, or other key parameters.
But from the options, notice that three points have the same y-coordinate (-4): (-11, -4), (-3, -4), and (2, -4), while one has a different y-coordinate (2, 8). This suggests the center of the hyperbola might be at (2, -4), and the foci would be along a horizontal or vertical line passing through this point.
For a standard hyperbola centered at (h, k), the foci are at (h ± c, k) for a horizontal hyperbola or (h, k ± c) for a vertical hyperbola, where c > a.
Given this, the only point far enough from (2, -4) to be a likely focus is (-11, -4), which is 13 units left of (2, -4). The other points are either too close or not aligned with the likely axis.
Therefore, the correct answer is:
A. (-11, -4
To determine which point is a focus of the hyperbola, we first need to understand the standard equation of a hyperbola and what it means to be a focus.
A hyperbola centered at ( h , k ) with a horizontal transverse axis is written as:
a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1
For a hyperbola with a vertical transverse axis, the equation is:
a 2 ( y − k ) 2 − b 2 ( x − h ) 2 = 1
The foci of a hyperbola are located c units away from the center along the transverse axis, where c 2 = a 2 + b 2 .
Now, the problem doesn't provide an explicit equation of the hyperbola or its center. However, if we assume that the given coordinates are possible candidates for the foci of a hyperbola, they generally must satisfy the condition of being c units away from the center along the transverse axis.
Given the options:
( − 11 , − 4 )
( − 3 , − 4 )
( 2 , − 4 )
( 2 , 8 )
It seems the hyperbola is horizontally aligned, seeing as the y-coordinate for options 1, 2, and 3 remain at − 4 .
For a horizontally aligned hyperbola with foci ( h ± c , k ) , the possible center might be around common points. Without additional details about the hyperbola's specific equation, we'll make an educated guess based on given points.
If we examine these points considering standard settings of a hyperbola, ( 2 , − 4 ) seems to be a good choice for a focus considering the coordinates provided suggest a symmetric distribution around the hypothetical center.
Therefore, the correct answer is:
O (2,-4)
